Primitive Complex Roots of Unity/Examples/Cube Roots
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Examples of Primitive Complex Roots of Unity
The primitive complex cube roots of unity are:
\(\ds \omega\) | \(=\) | \(\, \ds e^{2 \pi i / 3} \, \) | \(\, \ds = \, \) | \(\ds -\frac 1 2 + \frac {i \sqrt 3} 2\) | ||||||||||
\(\ds \omega^2\) | \(=\) | \(\, \ds e^{4 \pi i / 3} \, \) | \(\, \ds = \, \) | \(\ds -\frac 1 2 - \frac {i \sqrt 3} 2\) |
Proof
There are $3$ (complex) cube roots of unity:
- $1, \omega, \omega^2$
We have:
\(\ds \omega\) | \(=\) | \(\ds e^{2 \pi i / 3}\) | ||||||||||||
\(\ds \omega^2\) | \(=\) | \(\ds e^{4 \pi i / 3}\) | ||||||||||||
\(\ds \omega^3\) | \(=\) | \(\ds 1\) | Cube Roots of Unity |
Also:
\(\ds \omega^2\) | \(=\) | \(\ds e^{4 \pi i / 3}\) | ||||||||||||
\(\ds \paren {\omega^2}^2\) | \(=\) | \(\ds \omega^3 \times \omega\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 1 \times \omega\) | Cube Roots of Unity | |||||||||||
\(\ds \) | \(=\) | \(\ds \omega\) | ||||||||||||
\(\ds \paren {\omega^2}^3\) | \(=\) | \(\ds \paren {\omega^3}^2\) | Cube Roots of Unity | |||||||||||
\(\ds \) | \(=\) | \(\ds 1 \times 1\) | Cube Roots of Unity | |||||||||||
\(\ds \) | \(=\) | \(\ds 1\) |
Trivially, $1$ is not a primitive complex cube root of unity because you cannot make either $\omega$ or $\omega^2$ by multiplying $1$ by itself as many times as you like.
Hence the result.
$\blacksquare$
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): root of unity
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): root of unity